Problem: Determine how many solutions exist for the system of equations. ${6x+3y = -24}$ ${-12x+2y = -20}$
Convert both equations to slope-intercept form: ${6x+3y = -24}$ $6x{-6x} + 3y = -24{-6x}$ $3y = -24-6x$ $y = -8-2x$ ${y = -2x-8}$ ${-12x+2y = -20}$ $-12x{+12x} + 2y = -20{+12x}$ $2y = -20+12x$ $y = -10+6x$ ${y = 6x-10}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -2x-8}$ ${y = 6x-10}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.